Exceptional biases in counting primes over functions fields
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929263240806400 |
|---|---|
| author | Bailleul, Alexandre Devin, Lucile Keliher, Daniel Li, Wanlin |
| author_facet | Bailleul, Alexandre Devin, Lucile Keliher, Daniel Li, Wanlin |
| contents | We study how often exceptional configurations of irreducible polynomials over finite fields occur in the context of prime number races and Chebyshev's bias. In particular, we show that three types of biases, which we call "complete bias", "lower order bias" and "reversed bias", occur with probability going to zero among the family of all squarefree monic polynomials of a given degree in $\mathbb{F}_q[x]$ as $q$, a power of a fixed prime, goes to infinity. The bounds given improve on a previous result of Kowalski, who studied a similar question along particular $1$-parameter families of reducible polynomials. The tools used are the large sieve for Frobenius developed by Kowalski, an improvement of it due to Perret-Gentil and considerations from the theory of linear recurrence sequences and arithmetic geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_13665 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Exceptional biases in counting primes over functions fields Bailleul, Alexandre Devin, Lucile Keliher, Daniel Li, Wanlin Number Theory 11K38, 11T55, 11N45 We study how often exceptional configurations of irreducible polynomials over finite fields occur in the context of prime number races and Chebyshev's bias. In particular, we show that three types of biases, which we call "complete bias", "lower order bias" and "reversed bias", occur with probability going to zero among the family of all squarefree monic polynomials of a given degree in $\mathbb{F}_q[x]$ as $q$, a power of a fixed prime, goes to infinity. The bounds given improve on a previous result of Kowalski, who studied a similar question along particular $1$-parameter families of reducible polynomials. The tools used are the large sieve for Frobenius developed by Kowalski, an improvement of it due to Perret-Gentil and considerations from the theory of linear recurrence sequences and arithmetic geometry. |
| title | Exceptional biases in counting primes over functions fields |
| topic | Number Theory 11K38, 11T55, 11N45 |
| url | https://arxiv.org/abs/2302.13665 |